Toniolo and Linder, Equation (13)

Average Error: 53.84% → 41.16%

Time: 35.7s

Precision: binary64

Cost: 47300

Math

\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

↓

\[\begin{array}{l} t_1 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \sqrt[3]{U \cdot t}\\ t_3 := \left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(t_1 \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)\\ \mathbf{if}\;t_3 \leq 5 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{{t_2}^{2}} \cdot \sqrt{n \cdot t_2}\right)\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - -2 \cdot \frac{\ell}{\frac{-Om}{\ell}}\right) + t_1 \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \left(-\sqrt{\left(n \cdot U\right) \cdot \left(\frac{U*}{Om} \cdot \frac{n}{Om} + \frac{-2}{Om}\right)}\right)\right)\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt
  (*
   (* (* 2.0 n) U)
   (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))

↓

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (pow (/ l Om) 2.0)))
        (t_2 (cbrt (* U t)))
        (t_3
         (*
          (* U (* n -2.0))
          (+ (* t_1 (- U U*)) (- (* 2.0 (/ (* l l) Om)) t)))))
   (if (<= t_3 5e-302)
     (* (sqrt 2.0) (* (sqrt (pow t_2 2.0)) (sqrt (* n t_2))))
     (if (<= t_3 2e+306)
       (sqrt
        (*
         (* (* 2.0 n) U)
         (+ (- t (* -2.0 (/ l (/ (- Om) l)))) (* t_1 (- U* U)))))
       (*
        (sqrt 2.0)
        (*
         l
         (- (sqrt (* (* n U) (+ (* (/ U* Om) (/ n Om)) (/ -2.0 Om)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
}

↓

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * pow((l / Om), 2.0);
	double t_2 = cbrt((U * t));
	double t_3 = (U * (n * -2.0)) * ((t_1 * (U - U_42_)) + ((2.0 * ((l * l) / Om)) - t));
	double tmp;
	if (t_3 <= 5e-302) {
		tmp = sqrt(2.0) * (sqrt(pow(t_2, 2.0)) * sqrt((n * t_2)));
	} else if (t_3 <= 2e+306) {
		tmp = sqrt((((2.0 * n) * U) * ((t - (-2.0 * (l / (-Om / l)))) + (t_1 * (U_42_ - U)))));
	} else {
		tmp = sqrt(2.0) * (l * -sqrt(((n * U) * (((U_42_ / Om) * (n / Om)) + (-2.0 / Om)))));
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))));
}

↓

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * Math.pow((l / Om), 2.0);
	double t_2 = Math.cbrt((U * t));
	double t_3 = (U * (n * -2.0)) * ((t_1 * (U - U_42_)) + ((2.0 * ((l * l) / Om)) - t));
	double tmp;
	if (t_3 <= 5e-302) {
		tmp = Math.sqrt(2.0) * (Math.sqrt(Math.pow(t_2, 2.0)) * Math.sqrt((n * t_2)));
	} else if (t_3 <= 2e+306) {
		tmp = Math.sqrt((((2.0 * n) * U) * ((t - (-2.0 * (l / (-Om / l)))) + (t_1 * (U_42_ - U)))));
	} else {
		tmp = Math.sqrt(2.0) * (l * -Math.sqrt(((n * U) * (((U_42_ / Om) * (n / Om)) + (-2.0 / Om)))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
end

↓

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * (Float64(l / Om) ^ 2.0))
	t_2 = cbrt(Float64(U * t))
	t_3 = Float64(Float64(U * Float64(n * -2.0)) * Float64(Float64(t_1 * Float64(U - U_42_)) + Float64(Float64(2.0 * Float64(Float64(l * l) / Om)) - t)))
	tmp = 0.0
	if (t_3 <= 5e-302)
		tmp = Float64(sqrt(2.0) * Float64(sqrt((t_2 ^ 2.0)) * sqrt(Float64(n * t_2))));
	elseif (t_3 <= 2e+306)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(-2.0 * Float64(l / Float64(Float64(-Om) / l)))) + Float64(t_1 * Float64(U_42_ - U)))));
	else
		tmp = Float64(sqrt(2.0) * Float64(l * Float64(-sqrt(Float64(Float64(n * U) * Float64(Float64(Float64(U_42_ / Om) * Float64(n / Om)) + Float64(-2.0 / Om)))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(U * t), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(N[(U * N[(n * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 5e-302], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[Power[t$95$2, 2.0], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+306], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(-2.0 * N[(l / N[((-Om) / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * (-N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(N[(N[(U$42$ / Om), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 5.00000000000000033e-302

   1. Initial program 88.14

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Simplified 85.32

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{2}, n \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
      Proof

      [Start] 88.14	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 88.14	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-*l* [=>] 88.19	\[ \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
      *-commutative [=>] 88.19	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)\right)}} \]

   3. Taylor expanded in l around 0 67.99

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{n \cdot \left(t \cdot U\right)}} \]

   4. Applied egg-rr 69.5

      \[\leadsto \sqrt{2} \cdot \color{blue}{\left(\sqrt[3]{t \cdot U} \cdot \sqrt{n \cdot \sqrt[3]{t \cdot U}}\right)} \]

   5. Applied egg-rr 50.44

      \[\leadsto \sqrt{2} \cdot \left(\color{blue}{\sqrt{{\left(\sqrt[3]{t \cdot U}\right)}^{2}}} \cdot \sqrt{n \cdot \sqrt[3]{t \cdot U}}\right) \]

   if 5.00000000000000033e-302 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 2.00000000000000003e306

   1. Initial program 1.99

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Applied egg-rr 1.99

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(-\frac{\ell \cdot \ell}{-Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 2

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{-\ell}{\frac{-Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      Proof

      [Start] 1.99	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(-\frac{\ell \cdot \ell}{-Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-/l* [=>] 2	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(-\color{blue}{\frac{\ell}{\frac{-Om}{\ell}}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      distribute-neg-frac [=>] 2	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{-\ell}{\frac{-Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   if 2.00000000000000003e306 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))

   1. Initial program 99.83

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Applied egg-rr 99.83

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(-\frac{\ell \cdot \ell}{-Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 89.62

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{-\ell}{\frac{-Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      Proof

      [Start] 99.83	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(-\frac{\ell \cdot \ell}{-Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      associate-/l* [=>] 89.62	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(-\color{blue}{\frac{\ell}{\frac{-Om}{\ell}}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      distribute-neg-frac [=>] 89.62	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{-\ell}{\frac{-Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in U around 0 96.54

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)\right)}} \]

   5. Simplified 88.56

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right) - \frac{n}{\frac{\frac{Om}{\ell} \cdot \frac{Om}{\ell}}{U*}}\right)\right)}} \]
      Proof

      [Start] 96.54	\[ \sqrt{2 \cdot \left(n \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)\right)} \]
      associate-*r* [=>] 96.57	\[ \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(\left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)}} \]
      *-commutative [=>] 96.57	\[ \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
      associate-*r* [=>] 99.26	\[ \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]
      associate-*r* [<=] 99.23	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]
      +-commutative [=>] 99.23	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + -1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)} \]
      mul-1-neg [=>] 99.23	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \color{blue}{\left(-\frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)\right)} \]
      unsub-neg [=>] 99.23	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)} \]
      unpow2 [=>] 99.23	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)} \]
      associate-*r/ [<=] 99.23	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)} \]
      associate-*r* [=>] 99.23	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)} \]
      *-commutative [=>] 99.23	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\color{blue}{\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)} - \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)\right)} \]
      associate-/l* [=>] 99.33	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right) - \color{blue}{\frac{n}{\frac{{Om}^{2}}{{\ell}^{2} \cdot U*}}}\right)\right)} \]
      associate-/r* [=>] 99.3	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right) - \frac{n}{\color{blue}{\frac{\frac{{Om}^{2}}{{\ell}^{2}}}{U*}}}\right)\right)} \]

   6. Taylor expanded in l around -inf 85.94

      \[\leadsto \color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}\right)} \]

   7. Simplified 83.68

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(-\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{U*}{Om} \cdot \frac{n}{Om} + \frac{-2}{Om}\right)}\right)} \]
      Proof

      [Start] 85.94	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}\right) \]
      mul-1-neg [=>] 85.94	\[ \color{blue}{-\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}} \]
      associate-*l* [=>] 85.9	\[ -\color{blue}{\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}\right)} \]
      distribute-rgt-neg-in [=>] 85.9	\[ \color{blue}{\sqrt{2} \cdot \left(-\ell \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}\right)} \]
      *-commutative [=>] 85.9	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{n \cdot \color{blue}{\left(U \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}}\right) \]
      associate-*r* [=>] 86.14	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{\color{blue}{\left(n \cdot U\right) \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)}}\right) \]
      cancel-sign-sub-inv [=>] 86.14	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \color{blue}{\left(\frac{n \cdot U*}{{Om}^{2}} + \left(-2\right) \cdot \frac{1}{Om}\right)}}\right) \]
      *-commutative [=>] 86.14	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{\color{blue}{U* \cdot n}}{{Om}^{2}} + \left(-2\right) \cdot \frac{1}{Om}\right)}\right) \]
      unpow2 [=>] 86.14	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{U* \cdot n}{\color{blue}{Om \cdot Om}} + \left(-2\right) \cdot \frac{1}{Om}\right)}\right) \]
      times-frac [=>] 83.68	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\color{blue}{\frac{U*}{Om} \cdot \frac{n}{Om}} + \left(-2\right) \cdot \frac{1}{Om}\right)}\right) \]
      metadata-eval [=>] 83.68	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{U*}{Om} \cdot \frac{n}{Om} + \color{blue}{-2} \cdot \frac{1}{Om}\right)}\right) \]
      associate-*r/ [=>] 83.68	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{U*}{Om} \cdot \frac{n}{Om} + \color{blue}{\frac{-2 \cdot 1}{Om}}\right)}\right) \]
      metadata-eval [=>] 83.68	\[ \sqrt{2} \cdot \left(-\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{U*}{Om} \cdot \frac{n}{Om} + \frac{\color{blue}{-2}}{Om}\right)}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 41.16

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right) \leq 5 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{{\left(\sqrt[3]{U \cdot t}\right)}^{2}} \cdot \sqrt{n \cdot \sqrt[3]{U \cdot t}}\right)\\ \mathbf{elif}\;\left(U \cdot \left(n \cdot -2\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - -2 \cdot \frac{\ell}{\frac{-Om}{\ell}}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \left(-\sqrt{\left(n \cdot U\right) \cdot \left(\frac{U*}{Om} \cdot \frac{n}{Om} + \frac{-2}{Om}\right)}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	47.61%
Cost	20688

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{\left(n \cdot U\right) \cdot \left(\frac{U*}{Om} \cdot \frac{n}{Om} + \frac{-2}{Om}\right)}\\ \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \left(-t_2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot t_1 + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{+195}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} + \frac{-2}{Om}\right)}\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+256}:\\ \;\;\;\;{\left({\left(2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(t_1, -2, t\right)\right)\right)\right)}^{0.25}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot t_2\right)\\ \end{array} \]

Alternative 2
Error	47.54%
Cost	14728

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot U\right) \cdot \left(\frac{U*}{Om} \cdot \frac{n}{Om} + \frac{-2}{Om}\right)}\\ \mathbf{if}\;\ell \leq -3.2 \cdot 10^{+99}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \left(-t_1\right)\right)\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+197}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} + \frac{-2}{Om}\right)}\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+256}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot t_1\right)\\ \end{array} \]

Alternative 3
Error	47.17%
Cost	14412

\[\begin{array}{l} \mathbf{if}\;\ell \leq -9.5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\frac{n}{\frac{\frac{Om}{\ell} \cdot \frac{Om}{\ell}}{U*}} + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{U*}{Om} \cdot \frac{n}{Om} + \frac{-2}{Om}\right)}\right)\\ \end{array} \]

Alternative 4
Error	45.03%
Cost	14412

\[\begin{array}{l} t_1 := \sqrt{\left(n \cdot U\right) \cdot \left(\frac{U*}{Om} \cdot \frac{n}{Om} + \frac{-2}{Om}\right)}\\ \mathbf{if}\;\ell \leq -2.2 \cdot 10^{+91}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \left(-t_1\right)\right)\\ \mathbf{elif}\;\ell \leq 1.68 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\frac{n}{\frac{\frac{Om}{\ell} \cdot \frac{Om}{\ell}}{U*}} + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot t_1\right)\\ \end{array} \]

Alternative 5
Error	48.6%
Cost	8524

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.15 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\frac{n}{\frac{Om}{\ell}} \cdot \left(\ell \cdot \frac{U*}{Om}\right) + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \frac{\ell \cdot -2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \frac{n}{\frac{Om}{U \cdot \ell}}\right)\right)}\\ \end{array} \]

Alternative 6
Error	47.78%
Cost	8524

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.15 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.76 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\frac{n}{\frac{\frac{Om}{\ell} \cdot \frac{Om}{\ell}}{U*}} + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+50}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+200}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \frac{\ell \cdot -2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \frac{n}{\frac{Om}{U \cdot \ell}}\right)\right)}\\ \end{array} \]

Alternative 7
Error	49.15%
Cost	8392

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+91}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.12 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{n}{Om} \cdot \frac{\ell \cdot \left(\ell \cdot U*\right)}{Om} + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+198}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \frac{\ell \cdot -2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \frac{n}{\frac{Om}{U \cdot \ell}}\right)\right)}\\ \end{array} \]

Alternative 8
Error	51.15%
Cost	7888

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \ell \cdot \frac{\ell \cdot -2}{Om}\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -1.15 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -2.15 \cdot 10^{-196}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-169}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+198}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \frac{n}{\frac{Om}{U \cdot \ell}}\right)\right)}\\ \end{array} \]

Alternative 9
Error	57.34%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+100}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \frac{n}{\frac{Om}{\ell} \cdot \frac{1}{U \cdot \ell}}\right)}\\ \end{array} \]

Alternative 10
Error	54.37%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;U \leq -1.7 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;U \leq -1.3 \cdot 10^{-235}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 11
Error	56.91%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+100}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \frac{n}{\frac{Om}{U \cdot \ell}}\right)\right)}\\ \end{array} \]

Alternative 12
Error	53.41%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 13
Error	63.08%
Cost	7369

\[\begin{array}{l} \mathbf{if}\;\ell \leq 3.8 \cdot 10^{+33} \lor \neg \left(\ell \leq 4.7 \cdot 10^{+97}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{n}{Om}\right)\right)}\\ \end{array} \]

Alternative 14
Error	56.93%
Cost	7369

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5.8 \cdot 10^{+95} \lor \neg \left(\ell \leq 2.2 \cdot 10^{+33}\right):\\ \;\;\;\;\sqrt{\frac{n \cdot \ell}{\frac{Om}{U}} \cdot \left(\ell \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \]

Alternative 15
Error	57.1%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.9 \cdot 10^{+99}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.22 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{n \cdot \ell}{\frac{Om}{U}} \cdot \left(\ell \cdot -4\right)}\\ \end{array} \]

Alternative 16
Error	61.46%
Cost	7236

\[\begin{array}{l} \mathbf{if}\;\ell \leq 3.8 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(-4 \cdot \frac{\ell \cdot \left(U \cdot \ell\right)}{Om}\right)}\\ \end{array} \]

Alternative 17
Error	62.58%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;U* \leq -2.25 \cdot 10^{+45} \lor \neg \left(U* \leq -2.4 \cdot 10^{-217}\right):\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 18
Error	62.64%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;U* \leq 1.3 \cdot 10^{-212}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot t\right)}\\ \end{array} \]

Alternative 19
Error	62.63%
Cost	6848

\[\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)} \]

Reproduce

herbie shell --seed 2023089 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))